Prince began his career in 1948 as an assistant stage manager, under the auspice of Broadway director George Abbott. After stage managing such musicals as Call Me Madam and Wonderful Town, Prince aligned with Robert E. Griffith to produce the musicals The Pajama Game, Damn Yankees, New Girl in Town, West Side Story, Fiorello! and Tenderloin.

He began his directorial career when he took over for George Abbott on the musical A Family Affair. He continued to produce with such musicals as A Funny Thing Happened on the Way to the Forum; Fiddler on the Roof; Flora, The Red Menace; and She Loves Me, which he also directed.

On Broadway, he has directed the original productions of Cabaret, Zorba, Company, Follies, A Little Night Music, Pacific Overtures, On the Twentieth Century, Sweeney Todd, Evita, Merrily We Roll Along, The Phantom of the Opera, Kiss of the Spider Woman, Parade and Lovemusik.